BINARY BLOCK CODING 



531 



^^^""'^^^LL 



(1 + y)'''-'' dy 



(43) 



c [(1 + yy - IY+'[{1 + y)i + 1]"-+^' 

 and, without difficulty, 



<p,{2e, ^) = 2'' (^^^^ ~ ^^) =J^^- 1^^^ _ 3) • . • a - 2e + 1) 



The roots are 1, 3, • • • , 2e — 1, and from (32): 



,(e), ^ __e: X / (1 + .i-)'(l - xT^'-^^ 



G' 



^^ 2« 27rz Jr (f - 2e - 



2« 27rz Jr (^ - 2e - 1)(^ - 1)(^ - 3) • • • (^ - 2e + 1) 

 = 2-''(l + .t)(1 + xf - (1 - xfY = X + x'^' 

 In the case 7 = we need the result 



^e(2e + 1, I) = 2''"-' 



.e + l, 



\{k - 1) 



from (43). It is then tedious but straightforward to obtain from (35) 



v2e— 2r 



= 1 + a; 

 One expects to get 



= 2-^^-M[(l - x) + (1 + x)r^' - [(1 - aO - (1 + x)r^'\ 



2e+l 



y(7) 



G^^'Ct) = x^ + .r 



2e+l-7 



from (-11), but verification appears to be complicated. 



Case IV: e = 1, n = 2' - 1 (^ = 3, 4, • • •) 



The single error correcting codes of Hamming^ are included here. 

 (The examples for t = I, resp. t = 2, appear under Case II, resp. Case 

 III, above.) Since n is always odd the condition that (pi{n — 1, ^) = 

 2^ — ?i + 1 have an integer root is automatically satisfied. For 7=1 

 the generating function is 



* 



(1 _ ,-)f(i _ ^y-^ 



G''\x) 



-■[ 



d^ 



2TiJr (^ - n)(2^ - 7i + 1) 



(1 + xy - (1 + .r)^^""^^(l - .t)^^"+^^ 

 1 + 11 



from which we ha\'e 



'■"=rii.{(:)-(-)'-(ir^O} 



5 even 



